How the choice of the observable may influence the analysis of nonlinear dynamical systems

A great number of techniques developed for studying nonlinear dynamical systems start with the embedding, in a d-dimensional space, of a scalar time series, lying on an m-dimensional object, d > m. In general, the main results reached at are valid regardless of the observable chosen. In a number of practical situations, however, the choice of the observable does influence our ability to extract dynamical information from the embedded attractor. This may arise in standard problems in nonlinear dynamics such as model building, control theory and synchronization. To some degree, ease of success will thus depend on the choice of observable simply because it is related to the observability of the dynamics. Investigating the Rössler system, we show that the observability matrix is related to the map between the original phase space and the differential embedding induced by the observable. This paper investigates a form for the observability matrix for nonlinear system which is more general than the previous one used. The problem of controllability is also mentioned.